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Module PE.PAS.U15.5 Analysis of nonseries/parallel systems comprised of nonrepairable components
Module PE.PAS.U15.5
Analysis of non-series/parallel systems comprised
of non-repairable components
U15.1 Introduction
It is often the case that a logic diagram modeling a system of non-repairable components is neither parallel nor series, and as a result, cannot be analyzed using the techniques developed in module U14. Different techniques must be used as a result. To motivate the idea, consider the configuration illustrated in Fig. U15.1 [1], two substations connected by two lines.
Fig. 15.1: Substation Configuration
Substation 1 has three breakers protecting two circuits and as a result, is referred to as a “breaker and a half scheme.” From a reliability point of view, let’s consider that this system “works” if power is delivered through the lines to at least one of the buses and the scheme “fails” if power cannot be delivered to either bus. The components that may fail are breakers B1-B5. For each breaker, the failure mode of interest is its inadvertent opening.
We want to construct the logic diagram for this system. One observes that the following constitute failure modes for this system: (B1,B2), (B4,B5), (B4,B2,B3), (B5,B1,B3). We may represent these failure modes using the following logic diagram.
Fig. U15.2a: Substation Logic Diagram
The logic diagram of U15.2a is referred to as a bridge diagram and, as one observes, is in a configuration that is not directly amenable to our approach for reducing networks comprised of series/parallel subsystems.
Therefore, we resort to different methods, with the intent essentially being to decompose such non-series/parallel systems into series/parallel systems so that we may then apply our familiar techniques. These methods are as follows:
· Decomposition
· Delta-star transformation
· Cut set method
· Tie set method
· Connection matrix method
· Event trees
· Fault trees
U15.2 Decomposition
The decomposition approach is also called the conditional probability approach [2] and the factoring algorithm [3]. In this approach, we reduce the logic diagram sequentially into sub-structures that are connected in series/parallel and then recombine these substructures using conditional probability.
We can apply this to the diagram of Fig. U15.2a. The basic idea stems from the recognition that:
Or, in terms of Fig. U15.2:
Denoting, as usual, RS as the probability that the system works, we can write this as
RS=RS(given B3 works)RB3 +RS(given B3 fails)QB3
The reliabilities of the system, given that B3 works, and given that B3 fails, can be observed from inspecting Fig U15.2, so that the system reliability is:
RS={(1-QB4QB5)(1-QB1QB2)}RB3+{1-(1-RB4RB1)(1-RB5RB2)}QB3
U15.3 Delta-Star Transformation
Logic diagrams that have “delta” configurations may be transformed to logic diagrams containing “star” or “Y” configurations, often resulting in a simpler configuration that is amenable to series/parallel analysis [4]. This same idea is often used in basic circuit analysis, but one should be aware that the equations used to make the transformation are quite different.
To derive the equations for transforming a logical “delta” into a logical “star,” we take a terminal perspective of the two diagrams, as indicated in Fig. U15.3, so that the reliability between any two terminals of the delta configuration must be equal to the reliability between any two identical terminals of the star configuration. Application of this principle leads to the equivalencies shown in Fig. U15.4.
Fig. U15.3: Delta-Star Transformation
Fig. U15.4: Delta-Star Equivalencies
Equating the reliabilities of each pair of diagrams in Fig. U15.4 results in three equations that can then be solved for RA, RB, and RC. The result of this effort is [4]:
In Fig. U15.2b, we see, for example, how the system of Fig. U15.2a can be transformed.
Fig. U15.2b: Use of Delta-Star Transformation
U15.4 Cutset method
The basic idea of the cutset method is stated loosely as follows.
1. Identify all failure modes in terms of component sets such that,
2. for any one component set, there is no extra component in the set (meaning that the set no longer causes a failure if any one of the components does not fail), then
3. we compute the system failure probability as the probability of the union of all of the sets.
Note that item 3 indicates that we compute the system failure probability. This would be QS. The direct computation of RS is not appropriate in the cutset method, for reasons that we will see.
Definition: A cutset K is a set of components whose failure results in system failure. The removal of the corresponding set of blocks in the logic diagram interrupts the continuity between the input and output of the diagram [1]. Removal of all components in any cutset “disconnects” the “input” from the “output” in the logical diagram.
Definition: A minimal cutset C is a cutset where the set remaining after a removal of any of its elements is no longer a cutset. This definition means that all components of a minimal cutset must be failed to cause system failure.
Example: Find all cutsets K and minimal cutsets C for Fig. U15.5.
Fig. U15.5: Logic Diagram for Illustrating Cutset Identification
(1,2), (1,2,3), (1,2,4), (1,2,5), (1,2,6), (1,2,7), (1,2,8), (1,2,3,4), (1,2,3,5), (1,2,3,6), (1,2,3,7), (1,2,3,8), (1,2,4,5), (1,2,4,6), (1,2,4,7), (1,2,4,8), (1,2,5,6), (1,2,5,7), (1,2,5,8), (1,2,6,7), (1,2,6,8), (1,2,7,8), (1,2,3,4,5), (1,2,3,4,6), (1,2,3,4,7), (1,2,3,4,8), (1,2,3,5,6), (1,2,3,5,7), (1,2,3,5,8), (1,2,3,6,8), (1,2,3,7,8), (1,2,4,5,6), (1,2,4,5,7), (1,2,4,5,8), (1,2,4,6,7), (1,2,4,6,8), (1,2,4,7,8), (1,2,5,6,7), (1,2,5,6,8), (1,2,5,7,8), (1,2,6,7,8),
(7,8), (1,7,8), (2,7,8), (3,7,8), (4,7,8), (5,7,8), (6,7,8), (1,3,7,8), (1,4,7,8), (1,5,7,8), (1,6,7,8), (2,3,7,8), (2,4,7,8), (2,5,7,8), (2,6,7,8), (3,4,7,8), (3,5,7,8), (3,6,7,8), (4,5,7,8), (4,6,7,8), (5,6,7,8), (1,3,4,7,8), (1,3,5,7,8), (1,3,6,7,8), (1,4,5,7,8), (1,4,6,7,8), (1,5,6,7,8), (2,3,4,7,8), (2,3,5,7,8), (2,3,6,7,8), (2,4,5,7,8), (2,4,6,7,8), (2,5,6,7,8), (3,4,5,7,8), (3,4,6,7,8), (3,5,6,7,8), (4,5,6,7,8),
(2,3,5), (2,3,4,5), (2,3,5,6), (2,3,5,7), (2,3,5,8), (2,3,4,5,6), (2,3,4,5,7), (2,3,4,5,8), (2,3,5,6,7), (2,3,5,6,8),
(1,4,6), (1,3,4,6), (1,4,5,6), (1,4,6,7), (1,4,6,8), (1,3,4,5,6), (1,3,4,6,7), (1,3,4,6,8), (1,3,4,5,6), (1,4,5,6,7), (1,4,5,6,8), (1,3,4,6,7), (1,4,5,6,7), (1,3,4,6,8), (1,4,5,6,8),
(4,5,7), (1,4,5,7), (2,4,5,7), (3,4,5,7), (4,5,6,7), (1,3,4,5,7), (1,4,5,6,7), (2,4,5,6,7), (1,3,4,5,7), (3,4,5,6,7), (1,4,5,6,7), (3,4,5,6,7), (1,4,5,6,7), (2,4,5,6,7), (3,4,5,6,7),
(3,6,8), (1,3,6,8), (2,3,6,8), (3,4,6,8), (3,5,6,8), (1,3,4,6,8), (1,3,5,6,8), (2,3,4,6,8), (2,3,4,6,8), (3,4,5,6,8), (10)
(1,5,6,8), (1,3,5,6,8)
(3,4,5,6), (3,4,5,6,8)
(2,5,6,7)
plus all combinations of 6 elements, 8!/[6!(8-6)!]=28 combinations
plus all combinations of 7 elements, 8!/[7!(8-7)!]=8 combinations
(1,2,3,4,5,6,7,8)
There are 169 cutsets.
Cutset identification algorithm:
1. Identify the minimum number of branches which would be a cutset independent of the identity of the branches. This is NC=6.
2. Identify the minimal cutsets C, which are the ones above that are bold faced.
èAn informal way to think about how we did this is that a minimal cutset is any set of branches “cut” by a monotonically increasing or decreasing curve from left to right that completely breaks all paths from “input” to “output.”
3. For each minimal cutset, enumerate every set containing the minimal cutset that does not contain a cutset already enumerated, up to cardinality NC-1 (no need to enumerate cutsets of 6 or more).
4. Enumerate every set of cardinality NC, NC+1, …, N (these will all be cutsets)
Well, why are we interested in cutsets? We are not, really.
But we are interested in minimal cutsets. These would be, for our example, C1=(1,2), C2=(7,8), C3=(2,3,5), C4=(1,4,6), C5=(4,5,7), C6=(3,6,8), C7=(1,5,6,8), C8=(3,4,5,6), C9=(2,5,6,7).
Why are we interested in minimal cutsets?
Because the probability of system failure is given by the probability that at least one minimal cut fails, which is the probability that C1 fails or C2 fails or C3 fails or C4 fails or C5 fails or C6 fails or C7 fails or C8 fails or C9 fails, i.e.,
Qsys=P(Fsys)=P(FC1ÈFC2ÈFC3ÈFC4ÈFC5ÈFC6ÈFC7ÈFC8ÈFC9)
Note that evaluation of Qsys excludes all non-minimal cutsets. Why is this? Recall C1=(1,2), so (1,2,3) is a non-minimal cutset.
System failure occurs if 1 and 2 fail; the state of component 3 makes no difference. So
P(system failure due to failure of (1,2,3))=P(1 and 2 failing)
So getting just the probabilities of the minimal cutsets is sufficient.
We can develop a logical model for our system as in Fig. U15.6:
Fig. U15.6: Equivalent Cutset Logic Diagram
Now one is tempted here to evaluate the system reliability as:
Rsys=RC1RC2RC3RC4RC5RC6RC7RC8RC9
However, this does not work! Why not?
To find out why evaluation of system reliability Rsys does not work for a series connection of cutsets, let’s expand the series connection of cutsets so that it models the individual components.
To start with, look closely at a single cutset, say C1=(1,2).
How does failure of C1 occur? It occurs on failure of component 1 AND component 2, i.e., P(C1)=P(F1∩F2). This can be modeled logically as a parallel combination of components 1 and 2.
In fact, all cutset blocks can be modeled logically as a parallel combination of their constituent components. So we can expand the logic diagram of Fig. U15.6 to that of Fig. U15.7.
Fig. U15.7: Expanded Logic Diagram
One immediately observes repeated blocks, indicating dependencies between them, i.e., each block does not work or fail independently. This is the reason why we should not evaluate the reliability of this logic diagram using since it is based on probability evaluation of joint events, P(SC1∩SC2∩SC3∩SC4∩SC5∩SC6∩SC7∩SC8∩SC9), and therefore only applies if each block works or fails independently.
But what if we want to obtain the unreliability, QS?
Then, as we have said, we must evaluate the probability of the union of the events:
Qsys=P(Fsys)=P(FC1ÈFC2ÈFC3ÈFC4ÈFC5ÈFC6ÈFC7ÈFC8ÈFC9)
Do dependencies cause problems here?
To answer this question, let’s do a simpler case of evaluating just P(FC1ÈFC2ÈFC3), as illustrated in Fig. U15.8:
Fig. U15.8
Generalizing the rule for the union of two events, which is P(AÈB)=P(A)+P(B)-P(A∩B), we have for three events:
P(FC1ÈFC2ÈFC3)
=P(FC1)+P(FC2)+P(FC3) LINE 1
-{P(FC1∩FC2)+P(FC2∩FC3)+P(FC1∩FC3)} LINE 2
+P(FC1∩FC2∩FC3) LINE 3
In terms of components failure probabilities, this would be:
P(FC1ÈFC2ÈFC3)
=Q1Q2+Q7Q8+Q2Q3Q5 LINE 1
-{Q1Q2Q7Q8+Q7Q8Q2Q3Q5+Q1Q2Q3Q5} LINE 2
+Q1Q2Q7Q8Q3Q5 LINE 3
Note:
1. The last term in LINE 2, Q1Q2Q3Q5, came from:
P(FC1∩FC3)=P(FC1)´P(FC3| FC1)=Q1Q2´Q3Q5
2. The term in LINE 3, Q1Q2Q7Q8Q3Q5, came from:
P(FC1∩FC2∩FC3)=P((FC1∩FC2)∩FC3)=P(FC1∩FC2)´P(FC3|(FC1∩FC2))
=Q1Q2Q7Q8´Q3Q5
Clearly, we still have trouble with dependencies, although we were able to effectively deal with them.
We could analyze the entire C1-C9 logic diagram this way, with the help of the general formula for the union of multiple events [5]:
P(FC1ÈFC2ÈFC3…ÈFCn)=
where Zr is the rth summation in this sequence of summations, i.e.,
such that
P(FC1ÈFC2ÈFC3…ÈFCn)
But this would become rather tedious, since we would have to deal with each of the joint probabilities very carefully so as to appropriately screen the dependencies.
Instead, make 3 observations in reference to the above calculation.
1. The magnitudes of the quantities in the LINEs get smaller, i.e.,
§ |LINE 1|>|LINE 2|, i.e., Z1>Z2
§ |LINE 2|>|LINE 3|, i.e., Z2>Z3
and we can generalize to say that |LINE I|>|LINE J| when J>I, i.e., Zi>Zj when j>i.
2. Evaluation of P(FC1ÈFC2ÈFC3) using only LINE 1 (Z1) is a good approximation, as long as Qi’s are reasonably small. Therefore,
P(FC1ÈFC2ÈFC3…ÈFCn)»P(FC1)+P(FC2)+…+P(FCn)=Z1
This approximation is exact if cutset failures are mutually exclusive (occurrence of one cutset failure prohibits occurrence of other cutset failures), rarely the case in engineering systems.
3. |LINE 2|>|LINE I| for any I=3,…n. Since LINE 2>0, and since it is subtracted from LINE 1 (which is also >0), it must be the case that LINE 1> P(FC1ÈFC2ÈFC3…ÈFCn). Therefore, our approximation P(FC1ÈFC2ÈFC3…ÈFCn) is an UPPER BOUND to the actual probability.
In fact, observation 3 can be carried a little further. In [5], it is stated (with proof given in the references to [5]), that
P(FC1ÈFC2ÈFC3…ÈFCn)£Z1
P(FC1ÈFC2ÈFC3…ÈFCn)³Z1-Z2
P(FC1ÈFC2ÈFC3…ÈFCn) £ Z1-Z2+Z3
and so on. Fig. U15.9 illustrates this phenomenon.
Fig. U15.9: Impact of Additional Terms on
Failure Probability Estimation
One thought that comes to mind in studying Fig. U15.9 is that evaluation of Z1 and Z1-Z2 provides an upper and lower bound, respectively, on the failure probability P(FC1ÈFC2ÈFC3…ÈFCn). However, as we have observed, evaluation of Z2 requires that we address the issue of dependencies between cutsets. This is no fun. Besides, there is a better way.
U15.5 Tieset method
The basic idea of the tieset method is stated loosely as follows.
1. Identify all success modes in terms of component sets such that,
2. for any one component set, there is no extra component in the set (meaning that the set no longer causes a success if any one of the components fails), then
3. compute the system success probability as the probability of the union of all of the tiesets.
Note that item 3 indicates that we compute the system success probability. This would be RS. The direct computation of QS is not appropriate in the tieset method, for reasons that we will see.
Definition: A tieset V is a set of components whose success results in system success. The presence of the corresponding set of blocks in the logic diagram ensures the continuity between the “input” and “output” of the diagram, i.e., the presence of all components in any tieset “connects” the “input” to the “output” in the logical diagram.
Definition: A minimal tieset T is a tieset where the set remaining after a removal of any of its elements is no longer a tieset. This definition means that all components of a minimal tieset must succeed to cause system success.
Example: Find all tiesets V and minimal tiesets T for Fig. U15.5.
Fig. U15.5: Logic Diagram for Illustrating Tieset Identification
(1,3,7), (1,2,3,7),(1,3,4,7),(1,3,5,7),(1,3,6,7),(1,3,7,8),…
(1,5,8),(1,2,5,8),(1,3,5,8),(1,4,5,8),(1,5,6,8),(1,5,7,8),…
(2,4,8),(1,2,4,8),(2,3,4,8),(2,4,5,8),(2,4,6,8),(2,4,7,8),…
(2,6,7), (1,2,6,7),(2,3,6,7),(2,4,6,7),(2,5,6,7),(2,6,7,8),…
(1,3,4,6,8), (1,2,3,4,6,8),(1,3,4,5,6,8),(1,3,4,6,7,8) …
(2,3,4,5,7),(1,2,3,4,5,7)
(1,4,5,6,7),(1,2,4,5,6,7)
(2,3,5,6,8)
Again, we are not really interested in the tiesets but rather in the minimal tiesets. These would be, for our example, T1=(1,3,7), T2=(1,5,8), T3=(2,4,8), T4=(2,6,7), T5=(1,3,4,6,8), T6=(2,3,4,5,7), T7=(1,4,5,6,7), T8=(2,3,5,6,8).
Why are we interested in minimal tiesets?
Because the probability of system success is given by the probability that at least one minimal tieset succeeds, which is the probability that T1 succeeds or T2 succeeds or T3 succeeds or T4 succeeds or T5 succeeds or T6 succeeds or T7 succeeds or T8 succeeds, i.e.,
Rsys=P(Ssys)=P(ST1ÈST2ÈST3ÈST4ÈST5ÈST6ÈST7ÈST8ÈST9)
Note that evaluation of Rsys excludes all non-minimal tiesets. Why is this? Recall T1=(1,3,7), so (1,2,3,7) is a non-minimal tieset.